Ismail M., Koelink E. (editors) Theory and Applications of Special Functions

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THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

Theorem

5.4.

N+m+l-ks

x([

m-ks

Proof.

Set a

q in Corollary 4.3 and invoke Lemma 5.2 for the inner

sum with

Conclusion

The primary object of this paper has been the development of a-

Gaussian polynomials. In light of their natural partition-theoretic in-

terpretation (Proposition 2.7)) it is surprising that they have not been

studied previously. It seems extremely likely that Proposition 2.7 has

already suggested itself to many workers. The first thing one notices is

that for a-Gaussian polynomials there is no lovely product formula like

(1.8) only a less satisfying sum (Proposition 2.8) which reduces to (1.8)

when a

q. In addition, the symmetry identity (Andrews, 1976, p. 35,

eq. (3.3.2))

has no simple analog for a-Gaussian polynomials. It may well be that

these two deficits discouraged further investigation especially in light of

the fact that the definition of a-Gaussian polynomials contains a sum

that is not naturally terminating.

A secondary object of this paper has been the study of the polynomial

refinements of "a-generalizations" of Rogers-Ramanujan type identities.

Such studies almost always have in mind (or, at least, in the back of

their mind) the famous Borwein conjecture (Andrews, 1995). Namely, if

then each of An(q), B,(q) and Cn(q) has non-negative coefficients.

a-Gaussian Polynomials

It is not hard to show (Andrews, 1995, p. 491) that

Much is known about polynomials of this general nature. Indeed the

main theorem in (Andrews et al., 1987) shows that many such polyno-

mials must have non-negative coefficients.

However, the right-hand side of (5.3) is

not

covered by (Andrews et al.,

1987), but nonetheless, we see easily that it has non-negative coefficient

by inspection of the left-hand side of (5.3).

While the investigation of polynomial "a-generalizations" has not

here led to further information on the Borwein conjecture, it should

be pointed out that it has provided new insights on truncated Rogers-

Ramanujan identities, a topic treated from a wholly different viewpoint

in (Andrews, 1993).

References

Andrews,

(1974). Problem 74-12. SIAM Review, 16.

Andrews,

(1976). The Theory of Partitions. Addison-Wesley, Reading. (Reis-

sued: Cambridge University Press, Cambridge, 1984, 1998).

Andrews,

(1989). On the proofs of the Rogers-Ramanujan identities. In Stanton,

D.,

editor, q-Series and Partitions, volume 18 of IMA Volume in Math. and Its

Appl., pages 1-14. Springer, New York.

Andrews,

(1990). q-trinominial coefficients and Rogers-Ramanujan type iden-

tities. In Berndt et al., B., editor, Analytic Number Theory, Progr. Math., pages

1-11.

Birkhauser, Boston.

Andrews,

(1993). On Ramanujan's empirical calculation for the Rogers-Ramanujan

identities. In A tribute to Emil Grosswald: number theory and related analysis, vol-

ume 143 of Contemp. Math., pages 27-35. American Mathematical Society, Prov-

idence, RI.

Andrews,

(1995). On

conjecture of Peter Borwein.

Symbolic Computation,

20:487-501.

Andrews,

and Berkovich, A. (2002). The WP-Bailey tree and its implications.

London Math. Soc.

(2),

66(3):529-549.

Andrews,

E.,

Bressoud,

M., Baxter, R. J., Burge, W., Forrester, P. J., and

Viennot,

(1987). Partitions with prescribed hook differences. Europ.

Math.,

8:341-350.

THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

Bailey, W.

(1949). Identities of the Rogers-Ramanujan type. Proc. London Math.

SOC.

(2),

5O:l-10.

Bressoud,

M. (1981a). Solution to problem 74-12. SIAM Review, 23:lOl-104.

Bressoud,

M. (1981b). Some identities for terminating q-series. Math. Proc. Camb.

Phil. SOC., 89:211-223.

Paule, P. (1994). Short and easy computer proofs of the Rogers-Ramanujan identities

and identities of similar type. Electronic

Combin., 1:Research Paper 10, approx.

9 pp. (electronic).

Schur, I. (1917). Ein beitrag zur additiven zahlentheorie. In S.-B. Preuss. Akad.

Wiss., Phys.-Math. Klasse, pages 302-321. (Reprinted: Gesamm, Abhand., Vol.

2, Springer, Berlin, 1973, pp. 117-136).

Stembridge,

(1990). Hall-Littlewood functions, plane partitions, and the Rogers-

Ramanujan identities. Trans. Amer. Math. Soc., 319:469-498.

Warnaar, S.

(2002). Partial sum analogs of the Rogers-Ramanujan identities.

Comb. Th. (A), 99:143-161.

Watson, G.

(1929).

new proof of the Rogers-Ramanujan identities.

London

Math. Soc., 4:4-9.

Zeilberger (a.k.a.

B. Ekhad

and

S. Tre),

(1990).

purely verification proof of

the first Rogers-Ramanujan identity.

Comb. Th. (A), 54(2):309-311.

ON A GENERALIZED GAMMA

CONVOLUTION RELATED TO

THE q-CALCULUS

Christian Berg

Department

Mathematics

University of Copenhagen

Universitetsparken

DK-2100

DENMARK

berg@math.ku.dk

Abstract

We discuss a probability distribution

depending on a parameter

and determined by its moments n!/(q;

q)n.

The treatment

is purely analytical. The distribution has been discussed recently by

Bertoin, Biane and Yor in connection with a study of exponential func-

tional~ of Lkvy processes.

Keywords: q-calculus, infinitely divisible distribution

Introduction

In (Bertoin et al., 2004) Bertoin et al. studied the distribution

the exponential functional

qN'

dt,

where 0

is fixed and

(Nt,

0) is

standard Poisson process.

They found the density

i4(x),

0 and its Laplace and Mellin trans-

forms. They also showed that a simple construction from

leads to the

density

found by Askey, cf. (Askey, 1989), and having log-normal moments. The

notation in (1.2) is the standard notation from (Gasper and Rahman,

1990), see below.

2005

Springer Science+Business Media, Inc.

THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

The distribution

has also appeared in recent work of Cowan and

Chiu (Cowan and Chiu, 1994), Dumas et al. (Dumas et al., 2002) and

Pakes (Pakes, 1996).

The proofs in (Bertoin et al., 2004) rely on earlier work on exponential

functionals which use quite involved notions from the theory of stochastic

processes, see (Carmona et al., 1994; Carmona et al., 1997).

The purpose of this note is to give a self-contained analytic treatment

of the distribution

and its properties.

In Section

we define a convolution semigroup (Iq,t)t,O of probabil-

ities supported by [0, m[, and it is given in terms of the corresponding

Bernstein function

(s)

log(-s; q), with LBvy measure v on 10, m[

having the density

exp (-xqdn)

dx x

n=O

The function

log(-s; q), is a Stieltjes transform of a positive measure

which is given explicitly, and this permits us to determine the potential

kernel of

The measure

Iq,~

is a generalized Gamma convolution in the

sense of Thorin, cf. (Thorin, 197713; Thorin, 1977a). The moment se-

quence of

is shown to be n!/(q; q),, and the nth moment of

IqTt

is a

polynomial of degree n in t. We give a recursion formula for the coeffi-

cients of these polynomials. We establish that

has the density

treatment of the theory of generalized Gamma convolutions can be

found in Bondesson's monograph (Bondesson, 1992). The recent paper

(Biane et al., 2001) contains several examples of generalized Gamma

convolutions which are also distributions of exponential functionals of

LBvy processes.

We shall use the notation and terminology from the theory of basic

hypergeometric functions for which we refer the reader to the monograph

by Gasper and Rahman (Gasper and Rahman, 1990). We recall the q-

shifted factorials

and (z; q)o

1. Note that (z; q), is an entire function of z.

On a generalized Gamma convolution related to the q-calculus 63

For finitely many complex numbers zl,

22,

. . .

zp we use the abbrevi-

ation

(zl,z2,. zp; q),

(21; 4)n (22; 4)n. (~p; q)n.

The q-shifted factorial is defined for arbitrary complex index

and this is related to Jackson's function

defined by

In Section

we introduce the entire function

and use it to express the Mellin transform of

Iq.

the density

&ven in (1.2) can be written as the product convolution

and another related distribution, see Theorem 3.2 below. The

Mellin transform of the density

can be evaluated as a special case

of the Askey-Roy beta-integral given in (Askey and Roy, 1986) and in

particular we have, see also (Askey, 1989):

The value of (1.5) is an entire function of c and equals h(c)h(l-c)/(q; q),.

The following formulas about the q-exponential functions, cf. (Gasper

and Rahman, 1990), are important in the following:

qn(n-1)/2Zn

Eqb)

(-2; q),, z

(a; q)n

(1.7)

n=O

The analytic method

We recall that a function

10, m[

I+

[0, m[ is called completely

monotonic, if it is

and (-l)kcp(k)(~)

0 for

0,1,. . . .

By the Theorem of Bernstein completely monotonic functions have the

form

~(s)

e-sx dol(x),

(2.1)

THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

where

a non-negative measure on [0, m[. Clearly

cp(O+)

a([O,

m[).

The equation (2.1) expresses that

is the Laplace transform of the

measure a.

To establish that a probability q on [O,

oo[

is infinitely divisible, one

shall prove that its Laplace transform can be written

e-"" dq(x)

exp(-

(s)), s

where the non-negative function

has a completely monotonic deriv%

tive. If q is infinitely divisible, there exists a convolution semigroup

(qt)t>o of probabilities on [0, m[ such that q1

q and it is uniquely

determined by

cf. (Berg and Forst, 1975), (Bertoin, 1996). The function

is called the

Laplace exponent or Bernstein function of the semigroup. It has the

integral representation

where

0 and the LBvy measure

on 10,

oo[

satisfies the integrability

condition

x/(1+ x) dv(x)

oo.

is not identically zero the convo-

lution semigroup is transient with potential kernel

qt dt, and the

Laplace transform of

since

The generalized Gamma convolutions q are characterized among the

infinitely divisible distributions by the following property of the corre-

sponding Bernstein function

namely by

being a Stieltjes transform,

i.e. of the form

On a generalized Gamma convolution related to the q-calculus

where

0 and

is a non-negative measure on

[O,

XI[.

The relation

between

and

is that

This result was used in (Berg, 1981) to simplify the proof of a theo-

rem of Thorin (Thorin, 197713)) stating that the Pareto distribution is a

generalized Gamma convolution.

Theorem

2.1.

Let

be fixed. The function

f (s)

log(-s; q),

log(1

sqn),

(2.4)

n=O

is a Bernstein function. The corresponding convolution semigroup

((Iq,t)t,o)

consists of generalized Gamma convolutions and we have

The potential kernel

Iq,t

dt has the following completely mono-

tonic density

Xq(x)

pcpq(Y) dY.

(2.6)

where

is the continuous function

Proof.

The function

defined by (2.4) has the derivative

showing that

is a Stieltjes transform with

and

is the discrete

measure with mass

in each of the points

qen,

0. In particular

a Bernstein function with

and L6vy measure given by (1.3).

THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

Since

s x+s x

we get

([XI

denoting the integer part of x)

showing that

(s)/s is a Stieltjes transform. It follows by the Reuter-It6

Theorem, cf. (It6, 1974), (Reuter, 1956), (Berg, 1980), that

f(s) is a

Stieltjes transform. Since

is an increasing function mapping

]

oo[

onto the real line with

(0)

0 and f'(0)

1/(1-

q) we get

in the vague topology.

x ~]q-(~-l),q-~[, n

l,2,.

. .

we find

(log

(x; q),

inn)-'

These expressions define in fact a continuous function on [I,

m[,

vanish-

ing at the points q-n, n

0, so the measure

has the density

given

(2.7).

Using that the Stieltjes transformation is the second iteration

of the Laplace transformation, the assertion about the potential kernel

follows.

Denoting by

I,,

0 the exponential distribution with density

aexp(-ax) on the positive half-line, we have

so we can write

I,,J

as the infinite convolution

On a generalized Gamma convolution related to the q-calculus

If we let

ralt

denote the Gamma distribution with density

then we similarly have

Specializing (2.5) to

we have

and since the right-hand side of (2.8) is meromorphic in

with poles

-qWn, n

and in particular holomorphic for

1st

we know

that

has moments of any order with

cf. (Lukacs, 1960, p. 136). Here and in the following we denote by sn(p)

the nth moment of the measure p. However by (1.6) we have

hence

Since

has an analytic characteristic function, the corresponding

Hamburger moment problem is determinate. By Stirling's formula we

have

so also Carleman's criterion shows the determinacy, cf. (Akhiezer, 1965).

By (Berg, 1985, Cor. 3.3) follows that

Iq,t

is determinate for all

and by (Berg, 2000) the n'th moment

s,(I~,~)

is a polynomial of degree

n in

given by